Transcript UNIT 31

Unit 34
TRIGONOMETRIC FUNCTIONS
WITH RIGHT TRIANGLES
VARIATION OF FUNCTIONS


As the size of an angle increases, the sine,
tangent, and secant functions increase, but the
cofunctions (cosine, cotangent, cosecant)
decrease
Which is greater: cos 38° or cos 43°?


Since the cosine function decreases as the size of the
angle increases, cos 38° is greater than cos 43°
Which is greater: tan 42° or tan 24°?

Since the tangent function increases as the size of the
angle increases, tan 42° is greater than tan 24°
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FUNCTIONS OF
COMPLEMENTARY ANGLES


Two angles are complementary when their sum
is 90°. For example, 30° is the complement of
60°, and 60° is the complement of 30°
A function of an angle is equal to the cofunction
of the complement of the angle
sin A = cos (90° – A)
cos A = sin (90° – A)
tan A = cot (90° – A)
cot A = tan (90° – A)
sec A = csc (90° – A) csc A = sec (90° – A)
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FINDING UNKNOWN ANGLES

Procedure for determining an unknown
angle when two sides are given:
– In relation to the desired angle, identify two given
sides as adjacent, opposite, or hypotenuse
– Determine the functions that are ratios of the sides
identified in relation to the desired angle
– Choose one of the two functions, substitute the
given sides in the ratio
– Determine the angle that corresponds to the
quotient of the ratio
4
EXAMPLE OF FINDING AN ANGLE

Determine B to the nearest tenth of a
degree in the triangle below:
B
3.2"
5.7"
5
EXAMPLE (Cont)
– Since 5.7" is opposite B and 3.2" is adjacent B; either
the tangent or the cotangent function could be used
– Remember that when looking for an angle you will use
arctan in this case or tan-1 on your calculator
– Choosing the tangent function:
 5.7" 

tan B  
  60.7 Ans
 3.2" 
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FINDING UNKNOWN SIDES

Procedure for determining an unknown
side when an angle and a side are given:
– In relation to the given angle, identify the given side and
the unknown side as adjacent, opposite, or hypotenuse
– Determine the trigonometric functions that are ratios of
the sides identified in relation to the given angle
– Choose one of the two functions and substitute the
given side and given angle
– Solve as a proportion for the unknown side
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EXAMPLE OF FINDING A SIDE

Determine side x (to the nearest hundredth)
of the right triangle shown below:
x
46.3°
2.7 cm
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EXAMPLE (Cont)
– In relation to the 46.3° angle, the 2.7 cm side is the
adjacent side and side x is the hypotenuse. Thus, either
the cosine or the secant function could be used
– Choosing the cosine function:
cos 46.3 2.7 cm

1
x
so x = 3.91 cm Ans
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PRACTICE PROBLEMS
1.
2.
3.
4.
Which is greater: sin 48° or sin 32°?
Which is greater: csc 54.3° or csc 45.3°?
What is the cofunction of the complement of
sec 35°?
What is the cofunction of the complement of
cos 82°?
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PRACTICE PROBLEMS (Cont)
5.
Determine angle A to the nearest tenth of a
degree in the triangle shown below:
A
3.2"
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PRACTICE PROBLEMS (cont)
6.
7.
Determine side b in the triangle given below. Round
your answer to two decimal places.
Determine 1 (to the nearest tenth of a degree) in
the triangle given below.
1
3.25"
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PROBLEM ANSWER KEY
1.
2.
3.
4.
5.
6.
7.
sin 48°
csc 45.3°
csc 55°
sin 8°
46.7°
4.77 mm
33.7°
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